Bistatic inverse synthetic aperture radar imaging

ABSTRACT

A bistatic synthetic aperture radar (SAR) imaging system and method include: combining each radar return pulse from airborne radar platforms with a sinusoid; deskewing each reduced radar return pulse; estimating motion parameters based on a maximum likelihood estimation (MLE); performing MLE motion correction to generate motion-corrected radar return pulses; acquiring position and velocity estimates of the airborne radar platforms and scattering locations; defining bistatic range and velocity vectors; defining new bistatic range and velocity vectors in a new set of orthogonal axes; projecting vector distance differences between the radar scattering locations along the new set of orthogonal axes to generate new range and velocity measurements along the new set of orthogonal axes; converting the new range and velocity measurements to map Doppler frequency into cross-range; and forming a bistatic SAR image in range and cross-range based on cross-range extent derived from the Doppler frequency mapping.

BACKGROUND

Airborne maritime surveillance platforms employ low band radars thatoperate at UHF frequencies. Other low band surveillance radars have beendesigned to use S-Band or L-band. Typical platforms use radar with anapproximately 50 MHz operational bandwidth. If a 50 MHZ radar platformis used, the instantaneous pulse bandwidth is insufficient to achieve 1m resolution in down-range. Generally, a resolution of at least 1 m isrequired to carry out robust image exploitation of incoming targets. Therequired resolution can be obtained in cross-range with a bistatic radarimaging system that uses a precise oscillator to maintain coherencebetween the transmitting and receiving platforms.

There are other advantages to bistatic inverse synthetic aperture radar(ISAR) imaging. Tracking of the target from two (or more) platforms canbe combined to arrive at a more accurate kinematic profile (e.g.,position, speed, heading) of the target. There is reason to believe thatbistatic sea clutter may be less “spiky” than the equivalent monostaticsea clutter, and hence that bistatic geometries are more favorable forthe detection of small targets. Also, multipath mitigation may be moreeasily realizable with a bistatic imaging system due to higherdecorrelation of the multipath return in the radar compared to themonostatic scenario. In addition, the spatial diversity afforded bybistatic/multistatic systems allows for different aspects of a target tobe viewed simultaneously. Also, it may be increasingly difficult tosuccessfully focus jamming on multiple receivers in abistatic/multistatic system compared to a single receiver.

However, unlike the typical scenario in SAR imaging, airborne ISARimaging is more complicated because both the platform and the target aremoving during the course of the data collection dwell. Conventional ISARimaging techniques often produce images with unwanted distortion.

SUMMARY

The technology described herein relates to forming bistatic ISAR imagesof airborne targets by mapping Doppler frequency to cross-range, inphysical length units. Tracking of the target motion during datacollection enables the ISAR system to accurately motion compensate inthe airborne ISAR images. The implementations described herein formapping Doppler frequency to cross-range is robust under a variety oftracking errors and deviations from a broadside collection geometry.

One embodiment is a bistatic synthetic aperture radar (SAR) imagingmethod. The method includes receiving a plurality of radar return pulsesacquired by at least first and second airborne radar platforms, whereineach radar return pulse is generated in response to a correspondingtransmission pulse reflected from two or more radar scattering locationson a target. The method also includes combining each radar return pulsewith a sinusoid to reduce the radar return pulses to a base bandfrequency. The method also includes deskewing each reduced radar returnpulse to remove effects of its corresponding radar transmission pulse.The method also includes estimating motion parameters of the targetbased on maximum likelihood estimation (MLE) applied to the deskewedradar return pulses. The method also includes performing MLE motioncorrection to the deskewed radar return pulses based on the estimatedmotion parameters to generate motion corrected radar return pulses. Themethod also includes acquiring position and velocity estimates of thetwo or more airborne radar platforms and the one or more scatteringlocations on the target. The method also includes defining bistaticrange and velocity vectors based on the position and velocity estimatesof the first and second airborne radar platforms, the one or morescattering locations on the target, and the motion corrected radarreturn pulses. The method also includes defining new bistatic range andvelocity vectors by redefining the bistatic range and velocity vectorsin a new set of orthogonal axes. The method also includes projectingvector distance differences between the target radar scatteringlocations along the new set of orthogonal axes to generate new range andvelocity measurements along the new set of orthogonal axes. The methodalso includes converting the new range and velocity measurements inorder to map Doppler frequency into cross-range, measured in physicalunits of length. The method also includes forming a bistatic SAR imagein range and cross-range based on cross-range extent derived from theDoppler frequency mapping.

In some embodiments, the bistatic range and velocity vectors are definedaccording to:

${{\overset{\rightarrow}{p}}_{bistatic} = {\frac{\left( {{{\overset{\rightarrow}{p}}_{2}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}} \right)}{{{{\overset{\rightarrow}{p}}_{2}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}}} + \frac{\left( {{{\overset{\rightarrow}{p}}_{1}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}} \right)}{{{{\overset{\rightarrow}{p}}_{1}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}}}}},{and}$${\overset{\rightarrow}{p}}_{bistatic\_ vel} = \begin{bmatrix}{\frac{\left( {{{\overset{\rightarrow}{p}}_{2}^{\prime}(0)} - {{\overset{\rightarrow}{p}}_{s}^{\prime}(0)}} \right)}{{{{\overset{\rightarrow}{p}}_{2}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}}} + {{\frac{\left( {{{\overset{\rightarrow}{p}}_{2}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}} \right)}{{{{{\overset{\rightarrow}{p}}_{2}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}}}^{2}} \cdot \left( {{{\overset{\rightarrow}{p}}_{2}^{\prime}(0)} - {{\overset{\rightarrow}{p}}_{s}^{\prime}(0)}} \right)}\frac{\left( {{{\overset{\rightarrow}{p}}_{2}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}} \right)}{{{{\overset{\rightarrow}{p}}_{2}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}}}} +} \\{\frac{\left( {{{\overset{\rightarrow}{p}}_{1}^{\prime}(0)} - {{\overset{\rightarrow}{p}}_{s}^{\prime}(0)}} \right)}{{{{\overset{\rightarrow}{p}}_{1}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}}} + {{\frac{\left( {{{\overset{\rightarrow}{p}}_{1}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}} \right)}{{{{{\overset{\rightarrow}{p}}_{1}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}}}^{2}} \cdot \left( {{{\overset{\rightarrow}{p}}_{1}^{\prime}(0)} - {{\overset{\rightarrow}{p}}_{s}^{\prime}(0)}} \right)}\frac{\left( {{{\overset{\rightarrow}{p}}_{1}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}} \right)}{{{{\overset{\rightarrow}{p}}_{1}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}}}}}\end{bmatrix}$where {right arrow over (p)}_(bistatic) is the bistatic range vector and{right arrow over (p)}_(bistatic) _(_) _(vel) is the bistatic velocityvector, and {right arrow over (p)}₁(0) is a vector of an initialposition of the first airborne radar platform, {right arrow over(p)}₂(0) is a vector of an initial position of the second airborne radarplatform, {right arrow over (p)}₁′(0) is a vector of an initial velocityof the first airborne radar platform, {right arrow over (p)}₂′(0) is avector of an initial velocity of the second airborne radar platform and{right arrow over (p)}_(s) (0) is a vector of an initial position of ascatterer on the target, and {right arrow over (p)}′_(s)(0) is a vectorof an initial velocity of the scatterer on the target.

In some embodiments, the new bistatic range and velocity vectors areorthogonal parameters determined in accordance with:

${\overset{\rightarrow}{p}}_{bistatic},{and}$${\overset{\rightarrow}{p}}_{{bistatic\_ vel}{\_ new}} = {{\overset{\rightarrow}{p}}_{bistatic\_ vel} - {\frac{\left( {{\overset{\rightarrow}{p}}_{bistatic} \cdot {\overset{\rightarrow}{p}}_{bistatic\_ vel}} \right)}{{{\overset{\rightarrow}{p}}_{bistatic}}^{2}}{{\overset{\rightarrow}{p}}_{bistatic}.}}}$

In some embodiments, the new range and velocity measurements along thenew set of orthogonal axes are determined in accordance with:Δr _(bistatic) =−Δ{right arrow over (p)} _(s) ·{right arrow over (p)}_(bistatic)Δv _(bistatic) _(_) _(new) =Δ{right arrow over (p)} _(s) ·{right arrowover (p)} _(bistatic) _(_) _(vel) _(_) _(new)·,where Δ{right arrow over (p)}_(s) is difference between a dominatscattering location on the target and a second scattering location onthe target.

In some embodiments, the method includes using the new range andvelocity measurements to define cross-range resolution ΔR_(cross) _(_)_(range) and cross-range extent ΔR_(extent) in accordance with:

${{\Delta\; R_{cross\_ range}} = {\frac{1}{{\overset{\rightarrow}{p}}_{{bistatic\_ vel}{\_ new}}}\frac{\lambda}{T_{dwell}}}},{and}$${{\Delta\; R_{extent}} = {\frac{1}{{\overset{\rightarrow}{p}}_{{bistatic\_ vel}{\_ new}}}\frac{\lambda}{IPP}}},$where λ is the wavelength at the center frequency of the radar pulses,T_(dwell) is dwell duration of the radar pulses, and IPP is thereciprocal of pulse repetition frequency (PRF) of the radar pulses.

Another implementation is a bistatic synthetic aperture radar (SAR)imaging system. The system includes one or more processor and a memory.The memory includes executable code representing instructions that whenexecuted cause the system to receive a plurality of radar return pulsesacquired by at least first and second airborne radar platforms, whereineach radar return pulse is generated in response to a correspondingtransmission pulse reflected from two or more radar scattering locationson a target. The memory includes executable code representinginstructions that when executed cause the system to combine each radarreturn pulse with a sinusoid to reduce the radar return pulses to a baseband frequency. The memory includes executable code representinginstructions that when executed cause the system to deskew each reducedradar return pulse to remove effects of its corresponding radartransmission pulse. The memory includes executable code representinginstructions that when executed cause the system to estimate motionparameters of the target based on a maximum likelihood estimation (MLE)applied to the deskewed radar return pulses. The memory includesexecutable code representing instructions that when executed cause thesystem to perform MLE motion correction to the deskewed radar returnpulses based on the estimated motion parameters to generate motioncorrected radar return pulses. The memory includes executable coderepresenting instructions that when executed cause the system to acquireposition and velocity estimates of the two or more airborne radarplatforms and the one or more scattering locations on the target. Thememory includes executable code representing instructions that whenexecuted cause the system to define bistatic range and velocity vectorsbased on the position and velocity estimates of the first and secondairborne radar platforms, the one or more scattering locations on thetarget, and the motion corrected radar return pulses. The memoryincludes executable code representing instructions that when executedcause the system to define new bistatic range and velocity vectors byredefining the bistatic range and velocity vectors in a new set oforthogonal axes. The memory includes executable code representinginstructions that when executed cause the system to project vectordistance differences between the target radar scattering locations alongthe new set of orthogonal axes to generate new range and velocitymeasurements along the new set of orthogonal axes. The memory includesexecutable code representing instructions that when executed cause thesystem to convert the new range and velocity measurements in order tomap Doppler frequency into cross-range, measured in physical units oflength. The memory includes executable code representing instructionsthat when executed cause the system to form a bistatic SAR image inrange and cross-range based on cross-range extent derived from theDoppler frequency mapping.

In some embodiments, the bistatic range and velocity vectors are definedaccording to:

${\overset{\rightarrow}{p}}_{bistatic} = {\frac{\left( {{{\overset{\rightarrow}{p}}_{2}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}} \right)}{{{{\overset{\rightarrow}{p}}_{2}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}}} + \frac{\left( {{{\overset{\rightarrow}{p}}_{1}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}} \right)}{{{{\overset{\rightarrow}{p}}_{1}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}}}}$${\overset{\rightarrow}{p}}_{bistatic\_ vel} = \begin{bmatrix}{\frac{\left( {{{\overset{\rightarrow}{p}}_{2}^{\prime}(0)} - {{\overset{\rightarrow}{p}}_{s}^{\prime}(0)}} \right)}{{{{\overset{\rightarrow}{p}}_{2}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}}} + {{\frac{\left( {{{\overset{\rightarrow}{p}}_{2}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}} \right)}{{{{{\overset{\rightarrow}{p}}_{2}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}}}^{2}} \cdot \left( {{{\overset{\rightarrow}{p}}_{2}^{\prime}(0)} - {{\overset{\rightarrow}{p}}_{s}^{\prime}(0)}} \right)}\frac{\left( {{{\overset{\rightarrow}{p}}_{2}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}} \right)}{{{{\overset{\rightarrow}{p}}_{2}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}}}} +} \\{\frac{\left( {{{\overset{\rightarrow}{p}}_{1}^{\prime}(0)} - {{\overset{\rightarrow}{p}}_{s}^{\prime}(0)}} \right)}{{{{\overset{\rightarrow}{p}}_{1}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}}} + {{\frac{\left( {{{\overset{\rightarrow}{p}}_{1}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}} \right)}{{{{{\overset{\rightarrow}{p}}_{1}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}}}^{2}} \cdot \left( {{{\overset{\rightarrow}{p}}_{1}^{\prime}(0)} - {{\overset{\rightarrow}{p}}_{s}^{\prime}(0)}} \right)}\frac{\left( {{{\overset{\rightarrow}{p}}_{1}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}} \right)}{{{{\overset{\rightarrow}{p}}_{1}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}}}}}\end{bmatrix}$where {right arrow over (p)}_(bistatic) is the bistatic range vector and{right arrow over (p)}_(bistatic) _(_) _(vel) is the bistatic velocityvector, and {right arrow over (p)}₁(0) is a vector of an initialposition of the first airborne radar platform, {right arrow over(p)}₂(0) is a vector of an initial position of the second airborne radarplatform, {right arrow over (p)}₁′(0) is a vector of an initial velocityof the first airborne radar platform, {right arrow over (p)}₂′(0) is avector of an initial velocity of the second airborne radar platform and{right arrow over (p)}_(s)(0) is a vector of an initial position of ascatterer on the target, and {right arrow over (p)}′_(s)(0) is a vectorof an initial velocity of the scatterer on the target.

In some embodiments, the new bistatic range and velocity vectors areorthogonal parameters determined in accordance with:

${\overset{\rightarrow}{p}}_{bistatic},{and}$${\overset{\rightarrow}{p}}_{{bistatic\_ vel}{\_ new}} = {{\overset{\rightarrow}{p}}_{bistatic\_ vel} - {\frac{\left( {{\overset{\rightarrow}{p}}_{bistatic} \cdot {\overset{\rightarrow}{p}}_{bistatic\_ vel}} \right)}{{{\overset{\rightarrow}{p}}_{bistatic}}^{2}}{{\overset{\rightarrow}{p}}_{bistatic}.}}}$

In some embodiments, the new range and velocity measurements along thenew set of orthogonal axes are determined in accordance with:Δr _(bistatic) =−Δ{right arrow over (p)} _(s) ·{right arrow over (p)}_(bistatic)Δv _(bistatic) _(_) _(new) =Δ{right arrow over (p)} _(s) ·{right arrowover (p)} _(bistatic) _(_) _(vel) _(_) _(new)·,where Δ{right arrow over (p)}_(s) is difference between a dominantscattering location on the target and a second scattering location onthe target.

In some embodiments, the memory includes executable code representinginstructions that when executed cause the system to use the new rangeand velocity measurements to define cross-range resolution ΔR_(cross)_(_) _(range) and cross-range extent ΔR_(extent) in accordance with:

${{\Delta\; R_{cross\_ range}} = {\frac{1}{{\overset{\rightarrow}{p}}_{{bistatic\_ vel}{\_ new}}}\frac{\lambda}{T_{dwell}}}},{and}$${{\Delta\; R_{extent}} = {\frac{1}{{\overset{\rightarrow}{p}}_{{bistatic\_ vel}{\_ new}}}\frac{\lambda}{IPP}}},$where λ is the wavelength at the center frequency of the radar pulses,T_(dwell) is dwell duration of the radar pulses, and IPP is thereciprocal of pulse repetition frequency (PRF) of the radar pulses.

BRIEF DESCRIPTION OF THE DRAWINGS

The foregoing features of various embodiments of the invention will bemore readily understood by reference to the following detaileddescriptions in the accompanying drawings.

FIG. 1 is a schematic illustration of a bistatic imaging geometryenvironment of two platforms imaging a target, according to anillustrative embodiment.

FIG. 2 is a flowchart of a bistatic synthetic aperture radar (SAR)imaging method for processing radar pulses acquired with a radar,according to an illustrative embodiment.

FIG. 3 is a schematic illustration of a bistatic synthetic apertureradar (SAR) imaging system, according to an illustrative embodiment.

FIG. 4 is a flowchart of a bistatic synthetic aperture radar (SAR)imaging method, according to an illustrative embodiment.

DETAILED DESCRIPTION OF ILLUSTRATIVE EMBODIMENTS

The bistatic/multistatic ISAR methods and systems described herein(hereinafter “technology”) can provide one or more of the followingadvantages. One advantage of the technology is that accurate ISAR imagescan be generated because the method for mapping Doppler frequency tocross-range is robust under a variety of tracking errors and deviationsfrom a broadside collection geometry. Another advantage is thatexploitation of the details in an ISAR image is improved because thetechnology permits the system to accurately estimate cross-range.

FIG. 1 is a schematic illustration of a bistatic imaging geometryenvironment 100 of two airborne radar platforms 104 a and 104 b imaginga target 108, according to an illustrative embodiment. In thisembodiment, the target 108 has two distinct target scatterers 112 a and112 b. Two scatterers are not necessary for the final formation of anISAR image. In some embodiments, there is a single scatterer or three ormore scatterers. FIG. 1 illustrates the position vectors associated withthe platforms 104 a and 104 b, and the target 108 in the environment100. The description of the vectors is provided in Table 1.

TABLE 1 Postion Vectors Variable Definition {right arrow over (p)}₁Position of imaging platform 1 {right arrow over (p)}₂ Position ofimaging platform 2 {right arrow over (p)}_(s, 1) Position of a targetscatterer {right arrow over (p)}_(s, MLE) Position of dominant MLEscatterer{right arrow over (p)}₁ is a vector of the position of the firstairborne radar platform 104 a relative to a common reference point 116.{right arrow over (p)}₂ is a vector of the position of the secondairborne radar platform 104 b relative to the common reference point116. {right arrow over (p)}_(s,1) is a vector of the position of thefirst target scatterer 112 a of the target 108 relative to the commonreference point 116. {right arrow over (p)}_(s,MLE) is a vector of theposition of the second target scatterer 112 b of the target 108 relativeto the common reference point 116. In this embodiment, the second targetscatterer 112 b is the dominant scatterer of the target, denoted by MLE.

FIG. 2 is a flowchart 200 of a bistatic synthetic aperture radar (SAR)imaging method for processing radar pulses acquired with a pulse dopplerradar 204 applied to the environment 100 of FIG. 1. In one exemplaryembodiment, the pulse doppler radar 204 transmits a signal 202 andreceives a return signal 203 reflected from a target (e.g., the target108 of FIG. 1). The phase of the return signal is correlated with thephase of a model signal and due to doppler shifts in the return signal203, an SAR image is generated. Radar targets are frequentlymodeled/imaged by the superposition of a plurality of point scatterers.In this embodiment, the target 108 has two point scatterers 112 a and112 b.

The return signal 203 is passed through a low noise amplifier (LNA)(step 208) and then downconverted to reduce the radar return 203 pulsesto a baseband frequency. The radar return pulses 203 are downconvertedby combining each radar return 203 pulse with a sinusoid(e^(−2πif)RF^(t)) (step 210). The downconverted radar return pulses arethen provided to an A/D converter to convert the pulses to a digitalsignal for subsequent processing.

The baseband signal (output of A/D 212) is then passed through a matchedfilter (step 216) and then range deskewed (step 220) to remove theeffects of its corresponding radar transmission pulse. A Fouriertransform is applied at the application of the matched filter. Then,spectral equalization is achieved by factoring out the magnitude squareof the Fourier transform of the transmitted pulse to perform the rangedeskewing (step 220) of the signal.

The radar return output of the range deskew step (step 220) is passed toa maximum likelihood estimate (MLE) block 224 for extraction of thetarget 108 motion parameters. The residual line-of-sight motionparameters of the target are estimated up to the cubic term using an MLEalgorithm that operates on the most prominent target scatterer. The MLEestimate (step 224) determines the range signal, velocity signal, andacceleration signal for the dominant scatterer (scatterer 112 b inFIG. 1) in the field of view of the radar antenna 204. In thisembodiment, the assumption is made that the incoming target 108 isflying level and is not maneuvering. The MLE motion correction is thenapplied to produce a focused image (step 228). The correction is appliedas follows:Y(k,n)=W(k,n)exp(−jφ _(k,n)({circumflex over (r)},{circumflex over(v)},{circumflex over (α)})),

where W is the uncorrected signal, φ_(k,n) is the correction phase, and{circumflex over (r)}, {circumflex over (v)}, {circumflex over (α)} arethe correction parameters (location, velocity, acceleration,respectively) determined by the MLE algorithm.

After the focused image is produced (step 228) with respect to the MLEpoint (i.e., dominant target scatterer), the phasor of the secondprincipal scatterer is given by:

$\begin{matrix}{{Y\left( {k,n} \right)} = {z\;{\mathbb{e}}^{{- j}\frac{2\;\pi}{c}{({f_{RF} + {k\;\Delta\; f}})}{({{\Delta\; r} + {\Delta\;{vnIPP}}})}}}} & {{EQN}.\mspace{14mu} 1}\end{matrix}$where Δr is the range difference between the two scatterers, Δv is thevelocity difference between the two scatterers, c is the speed of light,f_(RF) is the waveform center frequency, Δf is the fast-time frequencystep, k and n are respectively the fast-time and slow-time indices, z isthe complex envelope of the scatterer, and IPP is the inter-pulseperiod. The range difference Δr is written as:

$\begin{matrix}\begin{matrix}{{\Delta\; r} = {{{{- \frac{\left( {{{\overset{\rightarrow}{p}}_{2}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}} \right)}{{{{\overset{\rightarrow}{p}}_{2}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}}}} \cdot \Delta}\;{\overset{\rightarrow}{p}}_{s}} - {{\frac{\left( {{{\overset{\rightarrow}{p}}_{1}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}} \right)}{{{{\overset{\rightarrow}{p}}_{1}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}}} \cdot \Delta}\;{\overset{\rightarrow}{p}}_{s}}}} \\{= {{- \Delta}\;{{\overset{\rightarrow}{p}}_{s} \cdot \left\lbrack {\frac{\left( {{{\overset{\rightarrow}{p}}_{2}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}} \right)}{{{{\overset{\rightarrow}{p}}_{2}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}}} + \frac{\left( {{{\overset{\rightarrow}{p}}_{1}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}} \right)}{{{{\overset{\rightarrow}{p}}_{1}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}}}} \right\rbrack}}}\end{matrix} & {{EQN}.\mspace{14mu} 2}\end{matrix}$where {right arrow over (p)}₁(0), {right arrow over (p)}₂(0), Δ{rightarrow over (p)}_(s) are the vectors defined in Table 1 in the middle ofthe data collection dwell. The position and velocity vector estimatesare acquired for the airborne radar platforms and the scatteringlocations on the target (step 232), where: Δ{right arrow over(p)}_(s)=({right arrow over (p)}_(s,MLE)−{right arrow over (p)}_(s,1)).The estimates can be acquired with a tracker. The target tracking iscarried out with methods and technologies common in the field, asdescribed in Samuel S. Blackman's book “Multiple-Target Tracking withRadar Applications”, Artech House, 1986, Dedham, Mass.”, the entirecontents of which is hereby incorporated by reference.

The velocity difference Δv is given by:

$\begin{matrix}{{\Delta\; v} = \begin{matrix}{\left\lbrack {\frac{\Delta\;{{\overset{\rightarrow}{p}}_{s} \cdot \left( {{{\overset{\rightarrow}{p}}_{2}^{\prime}(0)} - {{\overset{\rightarrow}{p}}_{s}^{\prime}(0)}} \right)}}{{{{\overset{\rightarrow}{p}}_{2}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}}} + {{\frac{\left( {{{\overset{\rightarrow}{p}}_{2}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}} \right)}{{{{{\overset{\rightarrow}{p}}_{2}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}}}^{2}} \cdot \left( {{{\overset{\rightarrow}{p}}_{2}^{\prime}(0)} - {{\overset{\rightarrow}{p}}_{s}^{\prime}(0)}} \right)}{\frac{\left( {{{\overset{\rightarrow}{p}}_{2}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}} \right)}{{{{\overset{\rightarrow}{p}}_{2}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}}} \cdot \Delta}\;{\overset{\rightarrow}{p}}_{s}}} \right\rbrack +} \\{\left\lbrack {\frac{\Delta\;{{\overset{\rightarrow}{p}}_{s} \cdot \left( {{{\overset{\rightarrow}{p}}_{1}^{\prime}(0)} - {{\overset{\rightarrow}{p}}_{s}^{\prime}(0)}} \right)}}{{{{\overset{\rightarrow}{p}}_{1}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}}} + {{\frac{\left( {{{\overset{\rightarrow}{p}}_{1}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}} \right)}{{{{{\overset{\rightarrow}{p}}_{1}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}}}^{2}} \cdot \left( {{{\overset{\rightarrow}{p}}_{1}^{\prime}(0)} - {{\overset{\rightarrow}{p}}_{s}^{\prime}(0)}} \right)}{\frac{\left( {{{\overset{\rightarrow}{p}}_{1}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}} \right)}{{{{\overset{\rightarrow}{p}}_{1}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}}} \cdot \Delta}\;{\overset{\rightarrow}{p}}_{s}}} \right\rbrack,}\end{matrix}} & {{EQN}.\mspace{14mu} 3}\end{matrix}$which can be written as:

$\begin{matrix}{{\Delta\; v} = {\Delta\;{{\overset{\rightarrow}{p}}_{s} \cdot \begin{bmatrix}{\frac{\left( {{{\overset{\rightarrow}{p}}_{2}^{\prime}(0)} - {{\overset{\rightarrow}{p}}_{s}^{\prime}(0)}} \right)}{{{{\overset{\rightarrow}{p}}_{2}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}}} + {{\frac{\left( {{{\overset{\rightarrow}{p}}_{2}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}} \right)}{{{{{\overset{\rightarrow}{p}}_{2}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}}}^{2}} \cdot \left( {{{\overset{\rightarrow}{p}}_{2}^{\prime}(0)} - {{\overset{\rightarrow}{p}}_{s}^{\prime}(0)}} \right)}\frac{\left( {{{\overset{\rightarrow}{p}}_{2}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}} \right)}{{{{\overset{\rightarrow}{p}}_{2}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}}}} +} \\{\frac{\left( {{{\overset{\rightarrow}{p}}_{1}^{\prime}(0)} - {{\overset{\rightarrow}{p}}_{s}^{\prime}(0)}} \right)}{{{{\overset{\rightarrow}{p}}_{1}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}}} + {{\frac{\left( {{{\overset{\rightarrow}{p}}_{1}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}} \right)}{{{{{\overset{\rightarrow}{p}}_{1}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}}}^{2}} \cdot \left( {{{\overset{\rightarrow}{p}}_{1}^{\prime}(0)} - {{\overset{\rightarrow}{p}}_{s}^{\prime}(0)}} \right)}\frac{\left( {{{\overset{\rightarrow}{p}}_{1}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}} \right)}{{{{\overset{\rightarrow}{p}}_{1}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}}}}}\end{bmatrix}}}} & {{EQN}.\mspace{14mu} 4}\end{matrix}$

The second term of EQN. 4 is defined as {right arrow over(p)}_(bistatic) _(_) _(vel), such that Δv then becomesΔv=Δ{right arrow over (p)} _(s) ·{right arrow over (p)} _(bistatic) _(_)_(vel).  EQN. 5

Now, the tracker measurements (step 232) of range and velocity for thetarget and the two imaging platforms is used to define the bistaticrange and bistatic velocity, and Δ{right arrow over (p)}_(s) is definedfrom its projections into two linearly independent vectors (step 236 ofFIG. 2) in accordance with:

$\begin{matrix}{{{\overset{\rightarrow}{p}}_{bistatic} = {\frac{\left( {{{\overset{\rightarrow}{p}}_{2}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}} \right)}{{{{\overset{\rightarrow}{p}}_{2}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}}} + \frac{\left( {{{\overset{\rightarrow}{p}}_{1}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}} \right)}{{{{\overset{\rightarrow}{p}}_{1}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}}}}},{and}} & {{EQN}.\mspace{14mu} 6} \\{{\overset{\rightarrow}{p}}_{{bistatic\_ ve}l} = \begin{bmatrix}{\frac{\left( {{{\overset{\rightarrow}{p}}_{2}^{\prime}(0)} - {{\overset{\rightarrow}{p}}_{s}^{\prime}(0)}} \right)}{{{{\overset{\rightarrow}{p}}_{2}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}}} + {{\frac{\left( {{{\overset{\rightarrow}{p}}_{2}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}} \right)}{{{{{\overset{\rightarrow}{p}}_{2}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}}}^{2}} \cdot \left( {{{\overset{\rightarrow}{p}}_{2}^{\prime}(0)} - {{\overset{\rightarrow}{p}}_{s}^{\prime}(0)}} \right)}\frac{\left( {{{\overset{\rightarrow}{p}}_{2}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}} \right)}{{{{\overset{\rightarrow}{p}}_{2}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}}}} +} \\{\frac{\left( {{{\overset{\rightarrow}{p}}_{1}^{\prime}(0)} - {{\overset{\rightarrow}{p}}_{s}^{\prime}(0)}} \right)}{{{{\overset{\rightarrow}{p}}_{1}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}}} + {{\frac{\left( {{{\overset{\rightarrow}{p}}_{1}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}} \right)}{{{{{\overset{\rightarrow}{p}}_{1}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}}}^{2}} \cdot \left( {{{\overset{\rightarrow}{p}}_{1}^{\prime}(0)} - {{\overset{\rightarrow}{p}}_{s}^{\prime}(0)}} \right)}\frac{\left( {{{\overset{\rightarrow}{p}}_{1}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}} \right)}{{{{\overset{\rightarrow}{p}}_{1}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}}}}}\end{bmatrix}} & {{EQN}.\mspace{14mu} 7}\end{matrix}$where {right arrow over (p)}_(bistatic) is the bistatic range vector and{right arrow over (p)}_(bistatic) _(_) _(vel) is the bistatic velocityvector, and {right arrow over (p)}₁(0) is a vector of an initialposition of the first airborne radar platform, {right arrow over(p)}₂(0) is a vector of an initial position of the second airborne radarplatform, {right arrow over (p)}₁′(0) is a vector of an initial velocityof the first airborne radar platform, {right arrow over (p)}₂′(0) is avector of an initial velocity of the second airborne radar platform andΔ{right arrow over (p)}_(s) is difference between a dominant scatteringlocation on the target and a second scattering location on the target.

The following two equations define the distance ∥Δ{right arrow over(p)}_(s)∥ of the MLE point from the other principal scatterer along theplane defined by the vectors {right arrow over (p)}_(bistatic) and{right arrow over (p)}_(bistatic) _(_) _(vel):Δrbistatic=−Δ{right arrow over (p)} _(s) ·{right arrow over (p)}_(bistatic)  EQN. 8,andΔv _(bistatic) =Δ{right arrow over (p)} _(s) ·{right arrow over (p)}_(bistatic) _(_) _(vel)·  EQN. 9

The distance of the MLE point from the other principal scatterers isapproximated from 2 linearly independent vectors. However, the ISAR mapof the target may be distorted by the non-orthogonality of the {rightarrow over (p)}_(bistatic) and {right arrow over (p)}_(bistatic) _(_)_(vel) vectors (this is the case when the data collection geometry isnot pure broadside, or 90° squint angle). It is more appropriate toredefine an orthogonal set of axes (step 240) defined by {right arrowover (p)}_(bistatic) and {right arrow over (p)}_(bistatic) _(_) _(vel)_(_) _(new), which involves a projection of the vector distancedifference between the target scatterers along the new set of orthogonalaxes in accordance with.

$\begin{matrix}{{\overset{\rightarrow}{p}}_{{bistatic\_ vel}{\_ new}} = {{\overset{\rightarrow}{p}}_{{bistatic\_ ve}l} - {\frac{\left( {{\overset{\rightarrow}{p}}_{bistatic} \cdot {\overset{\rightarrow}{p}}_{bistatic\_ vel}} \right)}{{{\overset{\rightarrow}{p}}_{bistatic}}^{2}}{{\overset{\rightarrow}{p}}_{bistatic}.}}}} & {{EQN}.\mspace{14mu} 10}\end{matrix}$

The new range and velocity measurements are defined in accordance with:Δr _(bistatic) =−Δ{right arrow over (p)} _(s) ·p _(bistatic)  EQN. 11Δv _(bistatic) _(_) _(new) =Δ{right arrow over (p)} _(s) ·{right arrowover (p)} _(bistatic) _(_) _(vel) _(_) _(new)·  EQN. 12The above redefinition is performed to avoid target distortions atnon-broadside geometries. From EQN. 12, we now have:

$\begin{matrix}{{\Delta\; v_{{bistatic}\;{\_ res}}} = {\left. \frac{\lambda}{T_{dwell}}\Rightarrow{\Delta\;{{\overset{\rightarrow}{p}}_{s} \cdot {\overset{\rightarrow}{p}}_{{bistatic\_ vel}{\_ new}}}} \right. = \frac{\lambda}{T_{dwell}}}} & {{EQN}.\mspace{14mu} 13}\end{matrix}$where λ is the wavelength at the center frequency of the radar pulses,T_(dwell) is dwell duration of the radar pulses, and IPP is thereciprocal of pulse repetition frequency (PRF) of the radar pulses.

The cross-range extent of the image is estimated and the Dopplerfrequency axis is mapped to cross-range in accordance with (step 248):

$\begin{matrix}{{{\Delta\; R_{cross\_ range}} = {\frac{1}{{\overset{\rightarrow}{p}}_{{bistatic\_ vel}{\_ new}}}\frac{\lambda}{T_{dwell}}}},{and}} & {{EQN}.\mspace{14mu} 14} \\{{\Delta\; R_{extent}} = {\frac{1}{{\overset{\rightarrow}{p}}_{{bistatic\_ vel}{\_ new}}}{\frac{\lambda}{IPP}.}}} & {{EQN}.\mspace{14mu} 15}\end{matrix}$

The final step is to form a bistatic ISAR image in range and cross-range(step 252). The image is formed by taking a 2-dimensional Fast FourierTransform of the complex time series given by EQN. 2. The Doppler extentin the image is converted to cross-range extent by applying the scalingfactor given in EQN. 14, where Δ/T_(dwell) is the Doppler frequency and1/∥{right arrow over (p)}_(bistatic) _(_) _(vel) _(_) _(new)∥ is thescaling factor that converts the Doppler frequency to cross-range.

FIG. 3 is a schematic illustration of a bistatic synthetic apertureradar (SAR) imaging system 300, according to an illustrative embodiment.The system 300 a SAR radar 302 (e.g., radar 202 of FIG. 2) whichincludes a radar antenna 304, a transmitter 308, and a receiver 312. Thesystem 300 also includes a controller 340. The controller 340 includes acommunication module 316, one or more input devices 320, one or moreoutput devices 328, one or more display devices 324, one or moreprocessors 332, and memory 336. The modules and devices described hereincan, for example, utilize the processor 332 to execute computerexecutable instructions and/or the modules and devices described hereincan, for example, include their own processor to execute computerexecutable instructions. It should be understood the controller 340 caninclude, for example, other modules, devices, and/or processors known inthe art and/or varieties of the described modules, devices, and/orprocessors.

The communication module 316 includes circuitry and code correspondingto computer instructions that enable the computing device tosend/receive signals to/from the antenna 304. For example, thecommunication module 316 provides commands from the processor 332 to thetransmitter 308 to control how the antenna 304 transmits radar pulsesduring operation. The communication module 316 also, for example,receives data corresponding to the radar return pulses received by thereceiver 312. The received data can be, for example, stored by thememory 336 or otherwise processed by the processor 332.

The input devices 320 receive information from a user (not shown) and/oranother computing system (not shown). The input devices 320 can include,for example, a keyboard, a scanner, a microphone, a stylus, a touchsensitive pad or display. The output devices 328 output informationassociated with the control module 120 (e.g., information to a printer,information to a speaker). The display devices 324 display operatinginformation and performance information (e.g., graphical representationsof information) regarding the SAR imaging methods. The processor 332executes the operating system and/or any other computer executableinstructions for the controller 340 (e.g., executes applications). Thememory 336 stores a variety of information/data, including profiles usedby the controller 340 to specify how the system 300 generates bistaticSAR images. The memory 336 can include, for example, long-term storage,such as a hard drive, a tape storage device, or flash memory; short-termstorage, such as a random access memory, or a graphics memory; and/orany other type of computer readable storage.

FIG. 4 is a flowchart 400 of a bistatic synthetic aperture radar (SAR)imaging method, according to an illustrative embodiment. The methodincludes transmitting a plurality of radar pulses towards a target (step404). The method also includes receiving a plurality of radar returnpulses acquired by at least first and second airborne radar platforms,wherein each radar return pulse is generated in response to acorresponding transmission pulse reflected from two or more radarscattering locations on a target (step 408). The method then includescombining each radar return pulse with a sinusoid to reduce the radarreturn pulses to a base band frequency (step 412), similarly asdescribed above with respect to step 210 of FIG. 2.

The method then includes deskewing each reduced radar return pulse toremove effects of its corresponding radar transmission pulse (step 416).The method then includes estimating motion parameters of the targetbased on a maximum likelihood estimation (MLE) applied to the deskewedradar return pulses (step 420).

The method then includes performing MLE motion correction to thedeskewed radar return pulses based on the estimated motion parameters togenerate motion corrected radar return pulses (step 424). The methodthen includes acquiring position and velocity estimates of the two ormore airborne radar platforms and the one or more scattering locationson the target (step 428). Step 428 can be performed by a tracker, asdescribed above with respect to, for example step 232 of FIG. 2. Themethod then includes defining bistatic range and velocity vectors basedon the position and velocity estimates of the first and second airborneradar platforms, the one or more scattering locations on the target, andthe motion corrected radar return pulses (step 432).

The method then includes defining new bistatic range and velocityvectors by redefining the bistatic range and velocity vectors in a newset of orthogonal axes (step 436). The method then includes projectingvector distance differences between the target radar scatteringlocations along the new set of orthogonal axes to generate new range andvelocity measurements along the new set of orthogonal axes (step 440).

The method then includes converting the new range and velocitymeasurements in order to map Doppler frequency into cross-range,measured in physical units of length (step 444). The method thenincludes forming a bistatic SAR image in range and cross-range based oncross-range extent derived from the Doppler frequency mapping (step448). The method then includes outputting the data associated with thebistatic SAR image (step 452). The output data can be stored in computermemory or processed further by, for example, a tracking system used totrack the target.

The above-described systems and methods can be implemented in digitalelectronic circuitry, in computer hardware, firmware, and/or software.The implementation can be as a computer program product that is tangiblyembodied in non-transitory memory device. The implementation can, forexample, be in a machine-readable storage device and/or in a propagatedsignal, for execution by, or to control the operation of, dataprocessing apparatus. The implementation can, for example, be aprogrammable processor, a computer, and/or multiple computers.

A computer program can be written in any form of programming language,including compiled and/or interpreted languages, and the computerprogram can be deployed in any form, including as a stand-alone programor as a subroutine, element, and/or other unit suitable for use in acomputing environment. A computer program can be deployed to be executedon one computer or on multiple computers at one site.

Method steps can be performed by one or more programmable processors, orone or more servers that include one or more processors, that execute acomputer program to perform functions of the disclosure by operating oninput data and generating output. Method steps can also be performed by,and an apparatus can be implemented as, special purpose logic circuitry.The circuitry can, for example, be a FPGA (field programmable gatearray) and/or an ASIC (application-specific integrated circuit).Modules, subroutines, and software agents can refer to portions of thecomputer program, the processor, the special circuitry, software, and/orhardware that implement that functionality.

Processors suitable for the execution of a computer program include, byway of example, both general and special purpose microprocessors, andany one or more processors of any kind of digital computer. Generally, aprocessor receives instructions and data from a read-only memory or arandom access memory or both. The essential elements of a computer are aprocessor for executing instructions and one or more memory devices forstoring instructions and data. Generally, a computer can be operativelycoupled to receive data from and/or transfer data to one or more massstorage devices for storing data. Magnetic disks, magneto-optical disks,or optical disks are examples of such storage devices.

Data transmission and instructions can occur over a communicationsnetwork. Information carriers suitable for embodying computer programinstructions and data include all forms of non-volatile memory,including by way of example semiconductor memory devices. Theinformation carriers can, for example, be EPROM, EEPROM, flash memorydevices, magnetic disks, internal hard disks, removable disks,magneto-optical disks, CD-ROM, and/or DVD-ROM disks. The processor andthe memory can be supplemented by, and/or incorporated in specialpurpose logic circuitry.

Comprise, include, and/or plural forms of each are open ended andinclude the listed parts and can include additional parts that are notlisted. And/or is open ended and includes one or more of the listedparts and combinations of the listed parts.

One skilled in the art will realize the invention may be embodied inother specific forms without departing from the spirit or essentialcharacteristics thereof. The foregoing embodiments are therefore to beconsidered in all respects illustrative rather than limiting of theinvention described herein. Scope of the invention is thus indicated bythe appended claims, rather than by the foregoing description, and allchanges that come within the meaning and range of equivalency of theclaims are therefore intended to be embraced therein.

The invention claimed is:
 1. A bistatic synthetic aperture radar (SAR)imaging method, the method comprising: receiving a plurality of radarreturn pulses acquired by at least first and second airborne radarplatforms, wherein each radar return pulse is generated in response to acorresponding transmission pulse reflected from two or more radarscattering locations on a target; combining each radar return pulse witha sinusoid to reduce the radar return pulses to a base band frequency;deskewing each reduced radar return pulse to remove effects of itscorresponding radar transmission pulse; estimating motion parameters ofthe target based on a maximum likelihood estimation (MLE) applied to thedeskewed radar return pulses; performing MLE motion correction to thedeskewed radar return pulses based on the estimated motion parameters togenerate motion corrected radar return pulses; acquiring position andvelocity estimates of the two or more airborne radar platforms and theone or more scattering locations on the target; defining bistatic rangeand velocity vectors based on the position and velocity estimates of thefirst and second airborne radar platforms, the one or more scatteringlocations on the target, and the motion corrected radar return pulses;defining new bistatic range and velocity vectors by redefining thebistatic range and velocity vectors in a new set of orthogonal axes;projecting vector distance differences between the target radarscattering locations along the new set of orthogonal axes to generatenew range and velocity measurements along the new set of orthogonalaxes; converting the new range and velocity measurements in order to mapDoppler frequency into cross-range, measured in physical units oflength; and forming a bistatic SAR image in range and cross-range basedon cross-range extent derived from the Doppler frequency mapping;wherein the steps in the method are carried out by a processor.
 2. Themethod of claim 1, wherein the bistatic range and velocity vectors aredefined according to: $\begin{matrix}{{\overset{\rightarrow}{p}}_{bistatic} = {\frac{\left( {{{\overset{\rightarrow}{p}}_{2}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}} \right)}{{{{\overset{\rightarrow}{p}}_{2}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}}} + \frac{\left( {{{\overset{\rightarrow}{p}}_{1}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}} \right)}{{{{\overset{\rightarrow}{p}}_{1}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}}}}} \\{{\overset{\rightarrow}{p}}_{bistatic\_ vel} = \begin{bmatrix}{\frac{\left( {{{\overset{\rightarrow}{p}}_{2}^{\prime}(0)} - {{\overset{\rightarrow}{p}}_{s}^{\prime}(0)}} \right)}{{{{\overset{\rightarrow}{p}}_{2}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}}} + {{\frac{\left( {{{\overset{\rightarrow}{p}}_{2}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}} \right)}{{{{{\overset{\rightarrow}{p}}_{2}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}}}^{2}} \cdot \left( {{{\overset{\rightarrow}{p}}_{2}^{\prime}(0)} - {{\overset{\rightarrow}{p}}_{s}^{\prime}(0)}} \right)}\frac{\left( {{{\overset{\rightarrow}{p}}_{2}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}} \right)}{{{{\overset{\rightarrow}{p}}_{2}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}}}} +} \\{\frac{\left( {{{\overset{\rightarrow}{p}}_{1}^{\prime}(0)} - {{\overset{\rightarrow}{p}}_{s}^{\prime}(0)}} \right)}{{{{\overset{\rightarrow}{p}}_{1}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}}} + {{\frac{\left( {{{\overset{\rightarrow}{p}}_{1}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}} \right)}{{{{{\overset{\rightarrow}{p}}_{1}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}}}^{2}} \cdot \left( {{{\overset{\rightarrow}{p}}_{1}^{\prime}(0)} - {{\overset{\rightarrow}{p}}_{s}^{\prime}(0)}} \right)}\frac{\left( {{{\overset{\rightarrow}{p}}_{1}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}} \right)}{{{{\overset{\rightarrow}{p}}_{1}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}}}}}\end{bmatrix}}\end{matrix}$ where {right arrow over (p)}_(multistatic) is the bistaticrange vector and {right arrow over (p)}_(bistatic) _(_) _(vel) is thebistatic velocity vector, and {right arrow over (p)}₁(0) is a vector ofan initial position of the first airborne radar platform, {right arrowover (p)}₂(0) is a vector of an initial position of the second airborneradar platform, {right arrow over (p)}₁′(0) is a vector of an initialvelocity of the first airborne radar platform, {right arrow over(p)}₂′(0) is a vector of an initial velocity of the second airborneradar platform and Δ{right arrow over (p)}_(s) is difference between adominant scattering location on the target and a second scatteringlocation on the target.
 3. The method of claim 2, wherein the newbistatic range and velocity vectors are orthogonal parameters determinedin accordance with:$\mspace{20mu}{{\overset{\rightarrow}{p}}_{bistatic},\mspace{20mu}{and}}$${\overset{\rightarrow}{p}}_{{bistatic\_ vel}{\_ new}} = {{\overset{\rightarrow}{p}}_{bistatic\_ vel} - {\frac{\left( {{\overset{\rightarrow}{p}}_{bistatic} \cdot {\overset{\rightarrow}{p}}_{bistatic\_ vel}} \right)}{{{\overset{\rightarrow}{p}}_{bistatic}}^{2}}{{\overset{\rightarrow}{p}}_{bistatic}.}}}$4. The method of claim 3, wherein the new range and velocitymeasurements along the new set of orthogonal axes are determined inaccordance with:Δr _(bistatic) =−Δ{right arrow over (p)} _(s) ·{right arrow over (p)}_(bistatic)Δv _(bistatic) _(_) _(new) =Δ{right arrow over (p)} _(s) ·{right arrowover (p)} _(bistatic) _(_) _(vel) _(_) _(new).
 5. The method of claim 4,comprising using the new range and velocity measurements to definecross-range resolution ΔR_(cross) _(_) _(range) and cross-range extentΔR_(extent) in accordance with:${{\Delta\; R_{cross\_ range}} = {\frac{1}{{\overset{\rightarrow}{p}}_{{bistatic\_ vel}{\_ new}}}\frac{\lambda}{T_{dwell}}}},{and}$${{\Delta\; R_{extent}} = {\frac{1}{{\overset{\rightarrow}{p}}_{{bistatic\_ vel}{\_ new}}}\frac{\lambda}{IPP}}},$where λ is the wavelength at the center frequency of the radar pulses,T_(dwell) is dwell duration of the radar pulses, and IPP is thereciprocal of pulse repetition frequency (PRF) of the radar pulses.
 6. Abistatic synthetic aperture radar (SAR) imaging system, the systemcomprising: one or more processor; and a memory, the memory includingexecutable code representing instructions that when executed cause thesystem to: receive a plurality of radar return pulses acquired by atleast first and second airborne radar platforms, wherein each radarreturn pulse is generated in response to a corresponding transmissionpulse reflected from two or more radar scattering locations on a target;combine each radar return pulse with a sinusoid to reduce the radarreturn pulses to a base band frequency; deskew each reduced radar returnpulse to remove effects of its corresponding radar transmission pulse;estimate motion parameters of the target based on a maximum likelihoodestimation (MLE) applied to the deskewed radar return pulses; performMLE motion correction to the deskewed radar return pulses based on theestimated motion parameters to generate motion corrected radar returnpulses; acquire position and velocity estimates of the two or moreairborne radar platforms and the one or more scattering locations on thetarget; define bistatic range and velocity vectors based on the positionand velocity estimates of the first and second airborne radar platforms,the one or more scattering locations on the target, and the motioncorrected radar return pulses; define new bistatic range and velocityvectors by redefining the bistatic range and velocity vectors in a newset of orthogonal axes; project vector distance differences between thetarget radar scattering locations along the new set of orthogonal axesto generate new range and velocity measurements along the new set oforthogonal axes; convert the new range and velocity measurements inorder to map Doppler frequency into cross-range, measured in physicalunits of length; and form a bistatic SAR image in range and cross-rangebased on cross-range extent derived from the Doppler frequency mapping.7. The system of claim 6, wherein the bistatic range and velocityvectors are defined according to:${\overset{\rightarrow}{p}}_{bistatic} = {\frac{\left( {{{\overset{\rightarrow}{p}}_{2}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}} \right)}{{{{\overset{\rightarrow}{p}}_{2}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}}} + \frac{\left( {{{\overset{\rightarrow}{p}}_{1}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}} \right)}{{{{\overset{\rightarrow}{p}}_{1}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}}}}$${\overset{\rightarrow}{p}}_{bistatic\_ vel} = \begin{bmatrix}{\frac{\left( {{{\overset{\rightarrow}{p}}_{2}^{\prime}(0)} - {{\overset{\rightarrow}{p}}_{s}^{\prime}(0)}} \right)}{{{{\overset{\rightarrow}{p}}_{2}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}}} + {{\frac{\left( {{{\overset{\rightarrow}{p}}_{2}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}} \right)}{{{{{\overset{\rightarrow}{p}}_{2}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}}}^{2}} \cdot \left( {{{\overset{\rightarrow}{p}}_{2}^{\prime}(0)} - {{\overset{\rightarrow}{p}}_{s}^{\prime}(0)}} \right)}\frac{\left( {{{\overset{\rightarrow}{p}}_{2}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}} \right)}{{{{\overset{\rightarrow}{p}}_{2}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}}}} +} \\{\frac{\left( {{{\overset{\rightarrow}{p}}_{1}^{\prime}(0)} - {{\overset{\rightarrow}{p}}_{s}^{\prime}(0)}} \right)}{{{{\overset{\rightarrow}{p}}_{1}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}}} + {{\frac{\left( {{{\overset{\rightarrow}{p}}_{1}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}} \right)}{{{{{\overset{\rightarrow}{p}}_{1}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}}}^{2}} \cdot \left( {{{\overset{\rightarrow}{p}}_{1}^{\prime}(0)} - {{\overset{\rightarrow}{p}}_{s}^{\prime}(0)}} \right)}\frac{\left( {{{\overset{\rightarrow}{p}}_{1}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}} \right)}{{{{\overset{\rightarrow}{p}}_{1}(0)} - {{\overset{\rightarrow}{p}}_{s}(0)}}}}}\end{bmatrix}$ where {right arrow over (p)}_(bistatic) is the bistaticrange vector and {right arrow over (p)}_(bistatic) _(_) _(vel) is thebistatic velocity vector, and {right arrow over (p)}₁(0) is a vector ofan initial position of the first airborne radar platform, {right arrowover (p)}₂(0) is a vector of an initial position of the second airborneradar platform, {right arrow over (p)}₁′(0) is a vector of an initialvelocity of the first airborne radar platform, {right arrow over(p)}₂′(0) is a vector of an initial velocity of the second airborneradar platform and Δ{right arrow over (p)}_(s) is difference between adominant scattering location on the target and a second scatteringlocation on the target.
 8. The system of claim 7, wherein the newbistatic range and velocity vectors are orthogonal parameters determinedin accordance with:$\mspace{20mu}{{\overset{\rightarrow}{p}}_{bistatic},\mspace{20mu}{and}}$${\overset{\rightarrow}{p}}_{{bistatic\_ vel}{\_ new}} = {{\overset{\rightarrow}{p}}_{bistatic\_ vel} - {\frac{\left( {{\overset{\rightarrow}{p}}_{bistatic} \cdot {\overset{\rightarrow}{p}}_{{bistatic\_ ve}l}} \right)}{{{\overset{\rightarrow}{p}}_{bistatic}}^{2}}{{\overset{\rightarrow}{p}}_{bistatic}.}}}$9. The system of claim 8, wherein the new range and velocitymeasurements along the new set of orthogonal axes are determined inaccordance with:Δr _(bistatic) =−Δ{right arrow over (p)} _(s) ·{right arrow over (p)}_(bistatic)Δv _(bistatic) _(_) _(new) =Δ{right arrow over (p)} ₂ ·{right arrow over(p)} _(bistatic) _(_) _(vel) _(_) _(new).
 10. The system of claim 9,wherein the memory includes executable code representing instructionsthat when executed cause the system to use the new range and velocitymeasurements to define cross-range resolution ΔR_(cross) _(_) _(range)and cross-range extent ΔR_(extent) in accordance with:${{\Delta\; R_{cross\_ range}} = {\frac{1}{{\overset{\rightarrow}{p}}_{{bistatic\_ vel}{\_ new}}}\frac{\lambda}{T_{dwell}}}},{and}$${{\Delta\; R_{extent}} = {\frac{1}{{\overset{\rightarrow}{p}}_{{bistatic\_ vel}{\_ new}}}\frac{\lambda}{IPP}}},$where λ is the wavelength at the center frequency of the radar pulses,T_(dwell) is dwell duration of the radar pulses, and IPP is thereciprocal of pulse repetition frequency (PRF) of the radar pulses.